Connectivity Graph-Codes

TL;DR

This paper introduces the concept of connectivity codes in graphs, characterizes their maximum size, and explores when this maximum is achieved in certain regular expanders.

Contribution

It defines connectivity codes, establishes an upper bound on their size, and identifies conditions under which this bound is tight, especially in regular expanders.

Findings

Maximum size of connectivity codes is at most 2^{edge-connectivity}

Equality holds for certain regular expanders

Equality does not hold for some common graph classes

Abstract

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$ of spanning subgraphs of $H$ a connectivity code for $H$ if the symmetric difference of any two distinct subgraphs in ${\cal G}$ is a connected spanning subgraph of $H$. It is easy to see that the maximum possible cardinality of such a collection is at most $2^{k'(H)} \leq 2^{\delta(H)}$, where $k'(H)$ is the edge-connectivity of $H$ and $\delta(H)$ is its minimum degree. We show that equality holds for any $d$-regular (mild) expander, and observe that equality does not hold in several natural examples including any large cubic graph, the square of a long cycle and products of a small clique with a long cycle.